introduction to particle physics lecture 13: the standard...

The Standard Model: Construction The Standard Model: Tests and status BSM motivation Supersymmetry

Introduction to particle physics

Lecture 13: The Standard Model

Frank Krauss

IPPP Durham

U Durham, Epiphany term 2010

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Outline

1 The Standard Model: Construction

2 The Standard Model: Tests and status

3 Why go beyond the Standard Model?

4 Supersymmetry

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The Standard Model

Matter sector

Spin-1/2 leptons and quarks in three generations (families):

In addition, a scalar (spin-0) Higgs boson as remainder ofspontaneous symmetry breaking of weak interactions

(not yet seen).

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Charges and interactions in the matter sector

Example: The first generation (u, d , νe and e):(The second and third generation are identical copies.)

particles weak charges e.m. charges

( (

u(r) u(b) u(g))

(

d (r) d (b) d (g))

)

Iu = | 12 ; + 12 〉

Id = | 12 ;− 12 〉

qu = 23

qd = − 13

(

νe

e−

)

Iν = | 12 ; + 12 〉

Ie = | 12 ;− 12 〉

qν = 0

qd = −1

Fermions arranged in weak isodoublets, all interacting weakly.(More precisely: left-handed iso-doublets with weak interactions and right-handed iso-singlets without them.)

Quarks come as colour-triplets, coupling to the strong force.Leptons do not carry a strong charge, therefore no strong interaction.

E.m. charges for Quarks and charged leptons → e.m. interactions;neutrinos do not interact electromagnetically (charge is 0).

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Interactions and generations

Strong, e.m., and neutral weak interactions do not change particletype (flavour); charged weak interactions do:In the lepton sector, generation number is conserved (lepton-typeconservation), in the quark sector it isn’t.

(No transitions between generations in lepton sector, in quark sector s → u and similar allowed in charged weak interactions.)

Masses in the matter sectorIn the Standard Model, neutrinos by definition massless.Other fermion masses:

1st generation 2nd generation 3rd generation

mu ≈ 5 MeV mc ≈ 1.5 GeV mt ≈ 175 GeVmd ≈ 5 MeV ms ≈ 200 MeV mb ≈ 4.5 GeVmνe

= 0MeV mνµ= 0MeV mντ

= 0MeV

me ≈ 0.511 MeV mµ ≈ 105 MeV mτ ≈ 1.75 GeV

Note: Recent experiments show that neutrinos mix.They also show that they have (tiny) masses, in the eV-range.A first hint for physics beyond the SM

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Quark mixing & CKM matrixCabibbo-Kobayashi-Maskawa

Inter-generation transitionsdominated by mass spectrumand CKM matrix:

Relative size of CKM Matrix (not to scale)

dominant: t → b, b → c , . . . .

Basic properties

Expand in small parameter Vus ≈ λ

Up to O(λ3):

VCKM =

0

B

B

B

@

1 −λ2

2λ Aλ3(ρ − iη)

λ 1 −λ2

2Aλ2

Aλ3(1 − ρ − iη) −Aλ2 1

1

C

C

C

A

Source of CP-violation in V13-elementsbut cosmologically not sufficient;

(Sakharov conditions)

unitarity of CKM matrix: triangles(VikV ∗

kj= δij );

size of CP-violation in SM given byarea of the triangle.

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“The” unitarity triangle

D.Hitlin, Talk at “Flavor in the Era of LHC”, 2005)

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Carriers of the interactionsConstruction principle of interactions in the Standard Model:

gauge invariance:

Invariance of the Lagrangian under global gauge transformationsensures conserved charges (electrical charge, weak isospin, colour),the demand for local gauge invariance generates the interactions,mediated by spin-1 gauge bosons (may carry charge themselves).Local gauge invariance ensures massless gauge bosons:mγ = mg = 0.Spontaneous symmetry breaking generates masses for the weakgauge bosons (mW ≈ 80.4 GeV, mZ ≈ 91.2 GeV).

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Spontaneous symmetry breaking: The Higgs boson

Basic idea: Have a symmetric theory with infinitely many, symmetricpotential ground states.

(Degenerate states of minimal energy as candidates for ground states, infinitely many number prevents tunneling.)

Pick one of them & expand fields around it (breaks symmetry).

Realisation in the Standard Model:

Add a complex scalar doublet, Φ;couple it to the weak gauge bosons of U(1)Y and SU(2)L;equip Φ with a non-trivial potential (“Mexican hat”) with infinitelymany minima 〈Φ0〉

2 = v2 (v = vacuum expectation value);

generates three massless “angular” modes (→ Goldstone bosons)and a massive “radial” one (→ Higgs boson);absorb the Goldstones in the gauge bosons, become massive (∝ v) –they now have 3 instead of 2 d.o.f.;massive remainder (the Higgs boson) couples to all other particlesand itself proportional to their mass.

Leads to a highly testable theory (some examples later).

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Feynman rules in the Standard Model

Fermion-fermion-gauge boson interactions (1st generation only):

W− (↓)

e−, d

W+ (↓)

νe, u

g

u, d

γ

e−, u, d

Z0

νe, e−, u, d

e−, u, d νe, e−, u, d u, d

νe, u e−, d

Important note: The charged bosons (W±) only allow forinter-generation mixing of the quarks (us, ub, dc, dt, cb, st).

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Gauge boson self-interactions (omitted the 4-boson vertices):

W+ W−

γ, Z0 g

gg

Interactions of the Higgs boson (omitted self-interactions):Interactions proportional to mass, therefore only heavy particles.

W±, Z0

H

W±, Z0

H

τ, b, t

τ, b, t

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The Standard Model: Tests and status

Precision tests of the Standard ModelThere are many relations between the parameters of the StandardModel, especially in the (weak) gauge sector:

For example, the masses of the bosons are directly related with theWeinberg angle by MZ = MW / cos θW ,also related with the Fermi-constant governing the muon decay,loop corrections in addition include quark masses, especially theheaviest one, the top quark. Therefore relations between, e.g. mW ,mt and mH .

These relations have very precisely been measured and calculated,the agreement is astonishing!

However, despite this apparent success, the building rests on theexistence of the Higgs mechanism to give a mass to the particles. Itsultimate test is the existence of the Higgs boson, not yet found.

Note also: Neutrino oscillations indicate that neutrinos have masses:a first sign that the Standard Model is not complete!

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Summary of precision tests

Right: Overall consistency(Fit of fundamental parameters from various data.)

Below: Consistency of mW /mt .(Higher order corrections by Higgs boson etc..)

80.3

80.4

80.5

150 175 200

mH [GeV]114 300 1000

mt [GeV]

mW

[G

eV]

68% CL

∆α

LEP1 and SLD

LEP2 and Tevatron (prel.)

March 2009

Measurement Fit |Omeas−Ofit|/σmeas

0 1 2 3

0 1 2 3

∆αhad(mZ)∆α(5) 0.02758 ± 0.00035 0.02767

mZ [GeV]mZ [GeV] 91.1875 ± 0.0021 91.1874

ΓZ [GeV]ΓZ [GeV] 2.4952 ± 0.0023 2.4959

σhad [nb]σ0 41.540 ± 0.037 41.478

RlRl 20.767 ± 0.025 20.742

AfbA0,l 0.01714 ± 0.00095 0.01643

Al(Pτ)Al(Pτ) 0.1465 ± 0.0032 0.1480

RbRb 0.21629 ± 0.00066 0.21579

RcRc 0.1721 ± 0.0030 0.1723

AfbA0,b 0.0992 ± 0.0016 0.1038

AfbA0,c 0.0707 ± 0.0035 0.0742

AbAb 0.923 ± 0.020 0.935

AcAc 0.670 ± 0.027 0.668

Al(SLD)Al(SLD) 0.1513 ± 0.0021 0.1480

sin2θeffsin2θlept(Qfb) 0.2324 ± 0.0012 0.2314

mW [GeV]mW [GeV] 80.399 ± 0.025 80.378

ΓW [GeV]ΓW [GeV] 2.098 ± 0.048 2.092

mt [GeV]mt [GeV] 173.1 ± 1.3 173.2

March 2009

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The quest for the Higgs boson: Basic findings

Higgs boson not yet found,therefore mH ≥ 114 GeV.

Precision data favour light mH

(see fit right)

Higgs boson couples to heavyobjects - all couplingsproportional to mass.

Problem: No heavy objectsinside typically used beams(e±, p).

0

1

2

3

4

5

6

10030 300

mH [GeV]∆χ

2

Excluded Preliminary

∆αhad =∆α(5)

0.02758±0.00035

0.02749±0.00012

incl. low Q2 data

Theory uncertaintyAugust 2009 mLimit = 157 GeV

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Why go beyond the Standard Model?

“Philosophical” motivation

SM is a model with 18(+1) parameters, can this be reduced?

Somewhat related: Can a GUT be constructed -a theory with only one interaction rather than three?

If there is a GUT, it presumably lives at scales O(1016GeV).A big desert from µEWSB to µGUT?(The “philosophical” hierarchy problem)

How can gravity be incorporated at all?Gauge constructions of gravity are tricky.

If dark matter is fundamental, where is it?The SM has no viable candidates.

Let’s not even start with dark energy/cosmological constant.

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Another nasty feature: The technical hierarchy problem

Consider two corrections to the mass of the Higgs boson:

∝ λHΛ2 ∝ −λ2t Λ

2

Each of them is quadratically divergent, with a brute-force cutoff Λ.(Think of it as limit of validity of SM, µGUT , or scale of new physics kicking in)

Remark: In QED, the fermion self-energy is only log-divergent due to gauge symmetry. Not a help here.

Huge fine-tuning of renormalisation mandatory to keep mH ≈ vev .(One-loop correction terms alone ∝ µ2

GUT)

Two solutions: Lower Λ (idea behind extra dimensions)or introduce a symmetry, e.g. λH = λ2

t (SUSY)

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Aside: Could the Standard Model survive up to µPlanck?

Remember: m2H = λv2

(v = vev = 246 GeV)

Two constraints on mass:1 Keep perturbativity:

λ → ∞ forbidden.2 Keep vacuum structure:

λ → 0 forbidden.

Therefore: “Stable island”in the middle

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Supersymmetry

What is supersymmetry?

Remember quantisation through operators:

Have creation and annihilation operators a(†): a

†|n〉 ∝ |n + 1〉,a|n〉 ∝ |n − 1〉, and a|0〉 = 0.Quantisation achieved through fixing their relationCommutator: [a, a

†] ∝ i , [a, a] = [a†, a†] = 0

Commutator for bosonic degrees of freedom.

Anticommutator {f1, f2} = f1f2 + f2f1 for fermionic d.o.f..

Supersymmetry:

Construct operation Q linking bosonic and fermionic states:Q|b〉 = |f 〉 & Q

†|f 〉 = |b〉.Demand invariance under this operationTherefore: For each bosonic d.o.f. in your model a fermionic one ismandatory and vice versa =⇒ b, f ∈ one “superfield”(This is the symmetry from above: Scalar and fermion belong to same superfield, therefore same coupling)

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The “philosophical” benefits of supersymmetry

Two “philosophical” in principle reasons:

1 The Coleman-Mandula Theorem statesthat the construction of a quantum theory of gravitation in form ofa local gauge theory is feasible only in the framework ofsupersymmetric theories.

2 The Haag-Sohnius-Lopuszanski Theorem statesthat the maximal symmetry of the S-matrix of a consistent QFT isgiven by the direct product of Lorentz-invariance, gauge symmetryand supersymmetry.

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Some “technological” motivation

Quadratic divergences are cancelled.For each loop with bosonic d.o.f. (sign = +), there is one withfermionic d.o.f. (sign = -) with exactly the same coupling, mass etc.:only difference is the sign!=⇒ Perfect cancellation of quadratic divergences.

Extra particles may help in enforcing unification of couplings.

The vacuum energy arising in second quantisation (zero-modeenergy of harmonic oscillator) is exactly cancelled by fermions=⇒ Vacuum energy is exactly 0

(Compare: Cosmological constant)

Typically, SUSY models have a natural dark matter candidate(a stable WIMP=LSP) with reasonable mass for CDM.

(Caveat: Only after SUSY-breaking)

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Field content before EWSB/SUSY breaking: all massless

Matter fields:left-handed doublets

right-handed singlets

Weyl-spinors/complex scalars

generations J = 1, 2, 3

(

uJ

dJ

)

L

, uJR , dJ

R

(

νJ

ℓJ

)

L

, ℓJR

(

uJ

dJ

)

L

, uJR , dJ

R

(

νJ

ℓJ

)

L

, ℓJR

Gauge fields:spin-1 bosons/Weyl-spinors

generators a = 1 . . . ng

G aµ, W±,0

µ, Bµ ψa

G , ψ±,0W , ψB

Higgs fields:2 doublets (i=1,2) of

Complex scalars/Weyl-spinors

(

H1i

H2i

)

L

(

ψ1Hi

ψ2Hi

)

L

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An unfortunate neccessity: Breaking SUSY

Pattern: SUSY partners with quantum numbers as SM particles,differing just in spin by a half unit

SUSY must be broken: no superpartner (with identical mass) found

Various mechanisms advocated, barely tractable

Way out: Breaking by hand through “soft term”(Terms that do not spoil the nice features, like absence of quadratic divergences)

This introduces ≈ 100 new parameters in MSSM:mostly boiling down to all possible mixings.

Typically imposed: R-parityPictorial: SUSY particles always pairwise in vertex!Consequence: A lightest stable SUSY particle (LSP).

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Summary

Construction of the Standard Model:Elementary particles and their properties, Feynman rules.

Some precision tests of the Standard Model, overwhelmingconsistency.

Motivation for going beyond the Standard Model.

Supersymmetry as an example.

To read: Coughlan, Dodd & Gripaios, “The ideas of particlephysics”, Sec 37, 38, 40, 42-44.

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introduction to particle physics lecture 13: the standard...

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